The Binomial Asset Pricing Model

The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory and probability theory. In the binomial asset pricing model, we model stock prices in discrete time, assuming that at each step, he stock price will change to of two possible values. Let $S_{0}$ be the initial stock price. There are two positive numbers, d and u with
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such that at the next period, the stock price will be either $dS_{0}$ or $uS_{0}.$ Typically, we take d and u to satisfy 0, so change of the stock price from $S_{0}$ to $dS_{0}$ represents a downward movement, and change of the stock price from $S_{0}$ to $uS_{0}$ represents an upward movement.

Binomial Model

We are tossing a coin, and when we get a Head, the stock price moves up, but when we get a Tail, the price moves down. We denote the price at time 1 by if the toss results in head (, and by $S_{1}(T)=dS_{0}$ if it results in tail (). After the second toss, the price will be one of:
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After three tosses, there are eight possible coin sequences. Assume that the third toss is the last one and denote by
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the set of all possible outcomes of the three tosses. The set $\Omega$ of all possible outcomes of a random experiment is called the sample space of the experiment. and the elements $\omega$ of $\Omega$ are called sample points. We note the k-th component of $\omega$ by $\omega _{k}.$ For example, when $\omega =HTH,$ we have and $\omega _{3}=H.$

The stock price S $_{k}$ at time k depends on the coin tosses. We write S $_{k}(\omega ).$

Example 1.

Set $S_{0}=4,u=2,$ and $d=\frac{1}{2}.$ Introduce the money market with interest . $ invested in the money market becomes $ in the next period.

Options Explained

You real estate developers know options. You buy an option on a plot of land for a tiny fraction of its total value. That gives you the right to buy it at a fixed price by a fixed date. If you get the financing, tenants, whatever you need, you can exercise the option and develop the property. Or, if the land goes up in value, you can simply sell the option at a profit to somebody else. Ultimately, if the deal never gets made, the option simply isn't exercised.

Insurance is also essentially an option of a different sort, an option that provides protection on your house and your car, your life. You pay a modest premium, and if something bad happens, say your house burns down, you get paid a much bigger amount to replace it. In essence then options are ways of coping with uncertainty over time, of transferring risk, of making contracts based on if this happens, if that happens, and so on. No wonder they're so popular.

The use of options in the stock market. So two final examples from Wall Street:

First, options can provide insurance on the stock market, just like in real life. Say you're two years away from retirement, with lots of stock in your pension account. You can't sell the stock. But you want to protect yourself against the market falling. Well, you can buy an option, cheaply, a so-called "put" option. What you buy is a contract, which is really just an insurance policy against your portfolio plummeting in value. It will pay you handsomely if the market falls within a certain time frame. On the other hand, suppose you're sure the market is going up in the next two years, but you can't afford or just don't want to buy a lot of stock. Instead, for far less money, you can buy an option, in this case a call option, which will pay you if the price rises by a certain date.

European Call option

Short Sales (sell)

A short sale is generally the sale of a stock you do not own. Investors who sell short believe the price of the stock will fall. If the price drops, you can buy the stock at the lower price and make a profit. If the price of the stock rises and you buy it back later at the higher price, you will incur a loss.

When you sell short, your brokerage firm loans you the stock. The stock you borrow comes from either the firm's own inventory, the margin account of another of the firm's clients, or another brokerage firm. As with buying stock on margin, your brokerage firm will charge you interest on the loan, and you are subject to the margin rules. If the stock you borrow pays a dividend, you must pay the dividend to the person or firm making the loan.

Long (buy)

One who has bought a stock to establish a market position and who has not yet closed out this position through an offsetting sale; the opposite of short (sell).

We assume be to the interest rate for both borrowing and lending. We also assume that
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This model would not make sense if we did not have this condition. For example, if $1+r\geq u,$ the rate of return on the money market is always at least as great and sometime greater than the return on the stock, and no one would invest in the stock. The inequality $d\geq 1+r$ cannot happen unless with r is negative or $d\geq 1.$ In the latter case, the stock does not really go "down" if we get a tail, it just goes up less than if we had gotten a head. One should borrow money at interest r and invest in the stock, since even in the worst case, the stock price rises at least at fast as the debt used to buy it.

Let us consider a European call option with strike price and expiration time 1. The option confers the right to buy the stock at time 1 for K dollars and so is worth S $_{1}-K$ at time 1 if $S_{1}-K$ is positive and is zero otherwise. Denote
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the value (payoff) of this option at expiration. Our first task is to compute the arbitrage price of this option at time zero.

Suppose at time zero you sell the call for $V_{0}$ dollars, where $V_{0}$ is still to be determined. You now have an obligation to pay off $(uS_{0}-K)^{+}$ if $\omega _{1}=H$ and to pay off $(dS_{0}-K)^{+}$ if $\omega _{1}=T.$ At the time you sell the option, you do not yet know which value $\omega _{1}$ will take. You hedge your short position in the option by buying shares of stock, where is still to be determined. you can use the proceeds $V_{0}$ of the sale of the option for this purpose, and then borrow if necessary at interest rate r to complete the purchase. If $V_{0}$ is more than necessary to buy the shares of stock, you invest the residual money at interest rate r. In either case, you will have shares of stock dollars invested in the money market, where this quantity might be negative. You will also own shares of stock. If the stock goes up, the value of your portfolio (excluding the short position in the option) is
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and you need to have $V_{1}(H).$ Thus, you want to choose $V_{0}$ and so that
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If the stock goes down, the value of your portfolio is
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and you need to have $V_{1}(T).$ Thus, you want to choose $V_{0}$ and so that
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There are two equations and two unknown, and we solve them
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and
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This is a discrete-time version of the famous "delta-hedging" formula for derivative securities.

The Delta factor approximates the change in the premium subsequent to a one unit change in the value of the underlying. The aim of the Delta hedge is to offset gains or losses on movements in the price of the underlying with losses or gains in the option premium. A decrease in the value of the underlying makes the put more valuable. An increase in the value of the underlying makes the put less valuable.
the number of shares of an underlying asset a hedge should hold is the derivative (in the sense of calculus) of the value of the derivative security w.r.t. the price of the underlying asset.
number of shares of stock one holds at time zero.
Note

This is the arbitrage price for the European call option with payoff $V_{1}$ at time 1. Define

So
- $\widetilde{p}$ and $\widetilde{q}$ are called risk-neutral probabilities.
Now consider a European call which pays off dollars at time . At expiration, the payoff of this option is

where $V_{2}$ and $S_{2}$ depend on $\omega _{1}$ and $\omega _{2},$ the first and second coin tosses. We want to determine the arbitrage price for this option at time zero. Suppose an agent sells the option at time zero for $V_{0}$ dollars, where $V_{0}$ is still to be determined. She then buys $V_{0}$ shares of stocks, investing dollars in the money market to finance this. At time 1, the agent has a portfolio value at

Or

and

After the first coin toss, the agent has $X_{1}$ dollars and can readjust her hedge. Suppose she decides to now hold shares of stock, where is allowed to depend on $\omega _{1}$ because the agent knows what value $\omega _{1}$ has taken. She invests the remainder of her wealth,

in the money market. In the next period, her wealth will be given by

She wants to have

Note $S_{2}$ and $V_{2}$ depend on $\omega _{1}$ and $\omega _{2},$ the outcomes of the first two coin tosses. Now we have six equations:

and

There are six unknowns, $X_{1}(H),$ and $X_{1}(T).$ To solve these equations, and thereby determine the arbitrage price $V_{0}$ at time zero of the option and the hedging portfolio

we begin with

Solving for

and we obtain the "delta-hedging" formula

and solve for

This equation gives the value the hedging portfolio should have at time 1 if the stock goes down between times 0 and 1. We define this quantity to be the arbitrary value of the option at time 1 if $\omega _{1}=T,$ and we denote it by $V_{1}(T),$ i.e.,

The hedger should choose her portfolio so that her wealth $X_{1}(T)$ if $\omega _{1}=T$ agrees with $V_{1}(T).$ Similarly,

and $X_{1}(H)=V_{1}(H),$ where V $_{1}(H)$ is the value of the option at time 1 if $\omega _{1}=H,$ denoted by

Finally,

and

become

and

Hence

and

Homework 2: Consider a European call which pays off dollars at time . Find $X_{1}(H),$ $X_{1}(T),$ $X_{2}(H),$ and $X_{2}(T).$ Due 1/31.

The Binomial Model

The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option.

At the end of the tree -- ie at expiration of the option -- all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values.

Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time. At the top of the tree you are left with one option price.

To get a feel for how the binomial model works you can use the on-line binomial tree calculators: either using the original Cox, Ross, & Rubinstein tree or the equal probabilities tree, which produces equally accurate results while overcoming some of the limitations of the C-R-R model. The calculators let you calculate European or American option prices and display graphically the tree structure used in the calculation. Dividends can be specified as being discrete or as an annual yield, and points at which early exercise is assumed for American options are highlighted.

Advantage:

The big advantage the binomial model has over the Black-Scholes model is that it can be used to accurately price American options. This is because with the binomial model it's possible to check at every point in an option's life (ie at every step of the binomial tree) for the possibility of early exercise (eg where, due to eg a dividend, or a put being deeply in the money the option price at that point is less than the its intrinsic value).

Where an early exercise point is found it is assumed that the option holder would elect to exercise, and the option price can be adjusted to equal the intrinsic value at that point. This then flows into the calculations higher up the tree and so on.

The on-line binomial tree graphical option calculator highlights those points in the tree structure where early exercise would have have caused an American price to differ from a European price.

The binomial model basically solves the same equation, using a computational procedure that the Black-Scholes model solves using an analytic approach and in doing so provides opportunities along the way to check for early exercise for American options.

Relationship to the Black-Scholes model:

The same underlying assumptions regarding stock prices underpin both the binomial and Black-Scholes models: that stock prices follow a stochastic process described by geometric brownian motion. As a result, for European options, the binomial model converges on the Black-Scholes formula as the number of binomial calculation steps increases. In fact the Black-Scholes model for European options is really a special case of the binomial model where the number of binomial steps is infinite. In other words, the binomial model provides discrete approximations to the continuous process underlying the Black-Scholes model.

Whilst the Cox, Ross & Rubinstein binomial model and the Black-Scholes model ultimately converge as the number of time steps gets infinitely large and the length of each step gets infinitesimally small this convergence, except for at-the-money options, is anything but smooth or uniform. To examine the way in which the two models converge see the on-line Black-Scholes/Binomial convergence analysis calculator. This lets you examine graphically how convergence changes as the number of steps in the binomial calculation increases as well as the impact on convergence of changes to the strike price, stock price, time to expiration, volatility and risk free interest rate.

My strategy evaluation model has a default of 100 steps which produces a good result at a good speed. Accuracy doesn't improve much for practical purposes over 200 steps but you can use more (eg 1,000 steps) if you want to but this will introduce annoying delays after entering each data field into the model.

Disadvantage:

As mentioned before the main disadvantage of the binomial model is its relatively slow speed. It's great for half a dozen calculations at a time but even with today's fastest PCs it's not a practical solution for the calculation of thousands of prices in a few seconds which is what's required for the production of the animated charts in my strategy evaluation model.

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Analysis

Black-Scholes price

4.8610

Range of steps considered 2 - 142

For steps in the range 139 - 142:

Maximum binomial price 4.8711

Average binomial price 4.8620

Minimum binomial price 4.8524

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